A coaxial cable compression connector includes a connector body having opposing first and second ends and a defined center passageway, an insulator disposed within the center passageway adjacent the first end of said connector body, and a compression sleeve movably connected to the second end of said connector body. In one version, the outer conductor of a prepared coaxial cable end and the center conductor are sequentially seized based on axial movement of the compression sleeve upon insertion of a prepared coaxial cable. The compression sleeve when axially moved from the second end to the first end sequentially enables a clamp to engage and seize each of the outer and center conductors. In another version, movement of the compression sleeve causes engagement of the center conductor of an engaged coaxial cable by a plurality of seizing elements disposed in relation to a conductive member into which the center conductor is advanced a predetermined distance.